Millennia Institute H2 Math papers, with video solutions

It is given that \(y=\sqrt{{{\mathrm{e}}^{x}}\cos x}\).

1. Show that \(2y\frac{\mathrm{d}y}{\mathrm{d}x}={{y}^{2}}-{{\mathrm{e}}^{x}}\sin x\).[2]

2. By further differentiation of the result in part (i), find the Maclaurin series for \(y\), up to and including the term in \({{x}^{2}}\).

   [4]

3. Using the standard series from the List of Formulae (MF 27). Expand \(\sqrt{{{\mathrm{e}}^{x}}\cos x}\) as far as the term in \({{x}^{2}}\) and verify that the same result is obtained in part (ii).

   [3]

A fruit stall sells \(2\) types of durians, XO and D24. For each type of durian, a randomly chosen durian has a mass that is normally distributed with mean and standard deviation, as well as unit price as shown in the following table.

1. Find the probability that the total mass of \(2\) randomly chosen XO durians is more than \(3\) times the mass of a randomly chosen D24 durian.[3]
2. Find the probability that the price of a randomly chosen XO durian differs from the price of a randomly chosen D24 durian by less than \(\$15\).[4]
3. A sample of \(n\) XO durians is randomly chosen. Find the least value of n such that the probability that the mean mass is more than \(2.25\) kg exceeds \(0.9\).[4]

Modelling assumption for this exercise: masses of all distinct selected durians are mutually independent and follow the stated distribution for their type.